Numerators and denominators

A fraction becomes 9/11, if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6 . Find the fraction.

Final Answer:

x =  7/9

Step-by-step explanation:

$$\begin{gathered} x=\frac{a}{b} \\ \frac{a+2}{b+2}=\frac{9}{11} \\ \frac{a+3}{b+3}=\frac{5}{6} \\ 11(a+2)=9(b+2) \\ 6(a+3)=5(b+3) \\ 11\cdot (a+2)=9\cdot (b+2) \\ 6\cdot (a+3)=5\cdot (b+3) \\ 11a-9b=-4 \\ 6a-5b=-3 \\ Row2-\frac{6}{11}\cdot Row1\to Row2 \\ -0.09b=-0.82 \\ b=\frac{-0.81818182}{-0.09090909}=9 \\ a=\frac{-4+9b}{11}=\frac{-4+9\cdot 9}{11}=7 \\ a=7 \\ b=9 \\ x=\frac{a}{b}=\frac{7}{9}=0.7778 \end{gathered}$$

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